Let be a Kummer -extension. Write . If we define and then it can be shown that . Therefore, the degree of the extension, , is . Define the embedding by . Then, . It turns out that (this notation means 'group generated by'). By the embedding it means . And this gives us a way to find the degree of a Kummer extension. What I do not see is why . I tried out some examples with and convinced myself that the only way to get an element when squared is when the element is a product of the -th roots. But I cannot find a nice proof to this.